Phase — Relative Alignment

Flow: Phase_RelativeAlignment_v1_0

Purpose:
Model phase as the relative alignment between repeating signals or within a single waveform, enabling structured teaching of phase relationships, timing offsets, and their impact on signal interaction and system behaviour.

System:
S = (I, R, B, K)

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Teaching Directive
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Execution_Mode:
Teaching

Audience:
Beginner to Intermediate learner

Teaching_Flow:
Explain → Visualise → Formalise → Apply

Instruction:
- Begin with intuitive explanation of phase as where a signal is within its cycle
- Explain that phase compares alignment between signals or points in time
- Introduce phase difference as a shift between waveforms
- Relate phase to timing offsets in signals such as sound waves or AC signals
- Reinforce with examples of in-phase and out-of-phase waveforms

Constraints:
H = ∅
Clarity before formalism

Goal:
Student understands that:
- phase describes position within a cycle
- phase difference describes alignment between signals
- phase is measured in degrees or radians
- phase affects how signals combine and interact

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State
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I = {
  signal_1,
  signal_2,
  phase_position,
  phase_difference,
  waveform_shape,
  time_state,
  reference_signal,
  cycle_fraction,
  measurement_units,
  system_response
}

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Rules
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R = {
  detect_signal_cycles,
  define_reference_point,
  measure_relative_alignment,
  compute_phase_difference,
  map_phase_to_cycle_fraction,
  interpret_signal_relationship,
  update_signal_characterisation
}

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Boundaries
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B = {
  signals_must_be_periodic_or_comparable,
  reference_signal_required,
  valid_time_or cycle_reference_required,
  phase_measurement_units_defined,
  no_unphysical_phase_without_cycle_definition
}

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Invariants
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K = {
  phase_represents_position_within_cycle,
  phase_difference_measures_relative_shift_between_signals,
  full_cycle_equals_360_degrees_or_2pi_radians,
  in_phase_means_zero_phase_difference,
  out_of_phase_means_nonzero_phase_difference,
  opposite_phase_means_180_degree_difference,
  phase_affects_signal_interaction_and_combination
}

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Execution
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E_t = expand(I, R)

V_t = {
  x ∈ E_t | x satisfies (B ∧ K)
}

Selection:
x* = argmax_(x ∈ V_t) P(x | I, R, B, K)

Update:
I_(t+1) = update(I_t, x*)

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Output
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O = quantified_signal_alignment_and_phase_relationship | (B, K)

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Failure Condition
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V_t = ∅ ⇒ O = FAIL

Meaning:
If no valid cycle reference, signal comparison, or alignment measure can be established, no coherent phase relationship can be evaluated.

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Interpretation
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Phase describes the position of a signal within its repeating cycle, or the relative alignment between two signals. When two signals are in phase, they reach their peaks and zero points at the same time. When they are out of phase, one signal is shifted in time relative to the other.

Phase difference is often measured in degrees or radians, with a full cycle corresponding to 360 degrees or 2π radians. A 180-degree phase difference means the signals are completely opposite, while smaller phase differences represent partial shifts.

Phase plays a critical role in signal interaction, including constructive and destructive interference, AC circuit behaviour, and communication systems.

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Core Relations
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Phase position = fraction_of_cycle
Full cycle = 360 degrees = 2pi radians

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Derived Relations
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Phase difference:
difference in alignment between two signals

In phase:
0 degrees difference

Out of phase:
non-zero difference

Opposite phase:
180 degrees difference

Wave shift:
time shift corresponds to phase shift

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Teaching Outcome
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Student can:
- Understand phase as position within a signal cycle
- Explain phase difference between signals
- Identify in-phase and out-of-phase relationships
- Relate phase to timing shifts
- Apply phase concepts to signal interaction and analysis